Qiaochu Yuan recommended Stevenhagen’s mathematical writings in general, so I did some additional searching. I found this page of lecture notes for algebraic number theory courses at Leiden University.
Stevenhagen’s notes on class field theory look particularly interesting. They start with particular examples, and explain what the theory means in those particular examples.
These notes by Stevenhagen provide an elementary introduction to rings of algebraic numbers.
The Geometry Center at the University of Minnesota was a pioneer in putting mathematics on the web. The Center specialized in visualization of advanced geometric topics.
The Center itself was closed in 1998, but their website is still available. The site is quite old (the pages that note that Netscape 2.0 is required are particularly poignant reminders), and many parts of it no longer work, but much of the content is still there.
Daniel Murfat has a nice series of notes on various mathematical topics, mostly algebraic geometry.
I came across this history of loops, a generalization of groups, where I picked up this interesting tidbit: loops are named after the Chicago Loop, the central business district of Chicago. (The elevated trains tracks form a loop that enclose it.)
The Economist has an article on the question of the hardest language to learn. They suggest that a language called Tuyuca is the answer. What makes Tuyuca unusual is that verbs carry an ending that indicates whether the statement is thought to be true or known to be true with certainty. Imagine a language with one tense for conjectures, and another for theorems.
The idea of a homomorphism extends neatly to general signatures. A function between two objects with the same signature is a homomorphism if it preserves all function and relation symbols. So φ is a homomorphism if for each n-ary function symbol f
φ( f(x1, …, xn) ) = f( φ(x1, …, φ(xn) )
and each n-ary relation symbol R
R(x1, …, xn) implies R(φ(x1, …, φ(xn))
This coincides with the usual definition of homomorphism for groups and rings. For partially-ordered sets, homomorphisms correspond to order-preserving maps.
This post at Geomblog is a nice survey of the different approaches to computational topology, which includes both computational approaches to topology and applications of topology to particular areas within computation.
Famously, Netflix offered a million dollar prize for a group that could beat Netflix’ own algorithm for predicting customer movie rankings. The contest, whose main beneficiary was Netflix itself, garnered considerable participation.
The Analytics X Competition is a similar contest, but aimed at benefiting the public directly. The contest is to predict crime rates by zip code in the city of Philadelphia. (The contest is privately funded, though, so the prizes are much smaller. Still it’s an interesting idea.)